S-wave conversion in marine surveys

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	13
Pages	485 - 496
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 13.1a

In a marine survey, the water depth is 100 m and a reflector is 3 km below the seafloor. Use Figure 13.1a to determine the optimum range of offsets for S-wave generation. Take Poisson’s ratio just below the seafloor as 0.35 and the P-wave velocity as 2.8 km/s. The velocity in sea water is 1.5 km/s.

Solution

Referring to Figure 13.1a(i), we see that the offset $x$ for the 3-km reflector is

${\begin{aligned}x=2(0.1\times \tan \theta _{1}+3\times \tan \delta _{2})=0.2\tan \theta _{1}+6\tan \delta _{2},\end{aligned}}$

(13.1a)

where $\theta _{1}$ is the angle of incidence of the P-wave at the bottom of the water layer, and $\delta _{2}$ is the angle of refraction of the converted S-wave.

To find $\delta _{2}$ for a given $\theta _{1}$, we need the S-wave velocity $\beta _{2}$. Using equation (1,8) in Table 2.2a, we find that $\beta /\alpha =0.48$ for $\sigma =0.35$, so $\beta _{2}=0.48\alpha _{2}=0.48\times 2.8=1.3\ {\rm {km/s}}$. To get the angles of incidence $\theta _{1}$ and of refraction $\delta _{2}$, we have from equation (3.1a),

${\begin{aligned}\sin \theta _{1}/1.5=\sin \delta _{2}/1.3\;,\;\delta _{2}={\rm {sin}}^{-1}\left(0.87\sin \theta _{1}\right).\end{aligned}}$

Figure 13.1a.  PSSP reflections generated by conversion at the sea floor (after Tatham and Stoffa, 1976). (i) Geometry; (ii) the conversion coefficient versus angle of incidence in the water; the curves are labeled with the seafloor P-wave velocity in km/s. [Note: Conversion coefficient = (amplitude of converted wave/amplitude of incident wave.)

The optimum angles of incidence $\theta _{1}$ for S-wave conversion for $\alpha =2.8\ {\rm {km/s}}$ are obtained by interpolating between the curves for $\alpha =2.5$ and 3.0 in Figure 13.1a(ii). This gives a range of about $38^{\circ }$ to $80^{\circ }$ for $\theta _{1}$. The corresponding values of $\delta _{2}$ are $32^{\circ }$ and $59^{\circ }$. Equation (13.1a) now gives offsets of 4 and 11 km for $\theta _{1}=38^{\circ }$ and $80^{\circ }$.

Problem 13.1b

Most marine S-wave surveys that wish to deal with S-waves utilize conversion at the reflector and recording with three-component geophones laid on the seafloor (OBC, ocean-bottom cables), so that only one mode conversion is involved. Assume a ray leaving an airgun source at $10^{\circ }$ to the vertical in water 100 m deep, an increase in P-wave velocities from 1.5 km/s at the seafloor to 3.0 km/s at a reflector 3 km below the sea floor (average velocity in the sediments of 2.25 km/s). Take $\sigma$ in the sediments as 0.3. What will be the source-geophone offset?

Solution

The P-wave gives an incident angle at the reflector of $\theta =\sin ^{-1}[(3.0/1.5)\sin 10^{\circ }]=20.3^{\circ }$. Using equation 1,8 in Table 2.2a, the S-wave velocity here is about 0.53 $\alpha$ or 1.59 km/s. Hence the angle of reflection is $\delta =\sin ^{-1}(0.53\sin 20.3^{\circ })=10.6^{\circ }$. The average direction of the P-wave ray in the sediments is $\sin ^{-1}[(2.25/1.5)\sin 10^{\circ }]=15.1^{\circ }$. With constant $\sigma$ in the sediments, $\beta /\alpha$ is constant and, hence, $\sin \delta /\sin \theta =0.53$. Hence, the average direction of the S-wave is $\sin ^{-1}(0.53\sin 15.1^{\circ }=7.9^{\circ })$. The offset is thus

${\begin{aligned}100\tan 10^{\circ }+3000\tan 15.1^{\circ }+3000\tan 7.9^{\circ }=1240\ {\rm {m}}.\end{aligned}}$

This is a very reasonable offset.

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Also in this chapter

• S-wave conversion in marine surveys
• Equally inclined orthogonal geophones
• Guided (channel) waves and normal-mode propagation
• Vertical seismic profiling
• Effect of velocity change on VSP traveltime
• Mapping the vertical flank of a salt dome
• Poission’s ratio from P- and S-wave traveltimes

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